Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{4p^2 + 28p}{9p + 72} \div \dfrac{p^2 + 6p - 7}{p + 8} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{4p^2 + 28p}{9p + 72} \times \dfrac{p + 8}{p^2 + 6p - 7} $ First factor the quadratic. $z = \dfrac{4p^2 + 28p}{9p + 72} \times \dfrac{p + 8}{(p + 7)(p - 1)} $ Then factor out any other terms. $z = \dfrac{4p(p + 7)}{9(p + 8)} \times \dfrac{p + 8}{(p + 7)(p - 1)} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ 4p(p + 7) \times (p + 8) } { 9(p + 8) \times (p + 7)(p - 1) } $ $z = \dfrac{ 4p(p + 7)(p + 8)}{ 9(p + 8)(p + 7)(p - 1)} $ Notice that $(p + 8)$ and $(p + 7)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ 4p\cancel{(p + 7)}(p + 8)}{ 9(p + 8)\cancel{(p + 7)}(p - 1)} $ We are dividing by $p + 7$ , so $p + 7 \neq 0$ Therefore, $p \neq -7$ $z = \dfrac{ 4p\cancel{(p + 7)}\cancel{(p + 8)}}{ 9\cancel{(p + 8)}\cancel{(p + 7)}(p - 1)} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $z = \dfrac{4p}{9(p - 1)} ; \space p \neq -7 ; \space p \neq -8 $